Maximum cuts: Improvements and local algorithmic analogues of the Edwards-Erdős inequality

Bylka, Stanisław; Idzik, Adam; Tuza, Źsolt

doi:10.1016/s0012-365x(98)00115-0

Cited by 13 publications

(12 citation statements)

References 11 publications

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“…Therefore, a graph always admits such a partition that can be found in polynomial time. On the other hand, those locally optimal cuts can be rather far from globally optimal ones (even when local vertex switching is allowed in a much wider sense), as proved in [21].…”

Section: Co-satisfactory Partitionmentioning

confidence: 95%

Satisfactory graph partition, variants, and generalizations

Bazgan

Tuza

Vanderpooten

2010

European Journal of Operational Research

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Section: Co-satisfactory Partitionmentioning

confidence: 95%

Satisfactory graph partition, variants, and generalizations

Bazgan

Tuza

Vanderpooten

2010

European Journal of Operational Research

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“…Classical guarantees include b(G) ≥ m 2 + 1 8 ( √ 8m + 1−1) (Edwards [5]) and, for connected graphs, b(G) ≥ m 2 + 1 4 (n − 1) (conjectured by Erdős [7] and proved by Edwards [6]). Bylka, Idzik, and Tuza [3] strengthened the Edwards-Erdős bound to b(G) ≥ m 2 + 1 2 2t−1 2t (n − 1) when G is connected and has no odd cycle of length less than 2t. Furthermore, this is sharp (using graphs whose blocks are (2t + 1)-cycles), and there is a linear-time algorithm to find a bipartite subgraph this big.…”

Section: Introductionmentioning

confidence: 93%

“…An elementary heuristic approach to finding large bipartite subgraphs is to start with an arbitrary vertex partition and then make local improvements. Bylka, Idzik, and Tuza [3] found a local switching algorithm that guarantees a cut as big as the bound of Edwards [5]. A simple type of local improvement is to move a vertex having more incident edges to vertices in its own set than in the other set; moving it to the other set increases the size of the cut.…”

Section: Introductionmentioning

confidence: 99%

Long Local Searches for Maximal Bipartite Subgraphs

Kaul¹,

West²

2008

SIAM J. Discrete Math.

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Given a partition of the vertices of a graph into two sets, a flip is a move of a vertex from its own set to the other, under the condition that it has more incident edges to vertices in its own set than in the other. Every sequence of flips eventually produces a bipartite subgraph capturing more than half of the edges in the graph. Each flip gains at least one edge. For an n-vertex loopless multigraph, we show that there is always a sequence of at most n/2 flips that cannot be extended, and we construct a graph having a sequence of 2 25 (n 2 + n − 31) flips.

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“…Other polynomial time Max-Cut approximation algorithms can be found in [2,3]. Because of the high proven performance guarantee of (GW), we focus on comparing our algorithm against it.…”

Section: Algorithm (Gw): the Goemans-williamson Algorithmmentioning

confidence: 99%

“…Let G ∈ G with n ≥ 2, let ε > 0, and let u be a minimizer of f + ε : V → R as in(2). Then u is not a constant function.Proof.…”

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confidence: 99%

A Max-Cut approximation using a graph based MBO scheme

Keetch¹,

Gennip²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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The Max-Cut problem is a well known combinatorial optimization problem. In this paper we describe a fast approximation method. Given a graph G, we want to find a cut whose size is maximal among all possible cuts. A cut is a partition of the vertex set of G into two disjoint subsets. For an unweighted graph, the size of the cut is the number of edges that have one vertex on either side of the partition; we also consider a weighted version of the problem where each edge contributes a nonnegative weight to the cut.We introduce the signless Ginzburg-Landau functional and prove that this functional Γ-converges to a Max-Cut objective functional. We approximately minimize this functional using a graph based signless Merriman-Bence-Osher scheme, which uses a signless Laplacian. We show experimentally that on some classes of graphs the resulting algorithm produces more accurate maximum cut approximations than the current state-of-the-art approximation algorithm. One of our methods of minimizing the functional results in an algorithm with a time complexity of O(|E|), where |E| is the total number of edges on G.

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Maximum cuts: Improvements and local algorithmic analogues of the Edwards-Erdős inequality

Cited by 13 publications

References 11 publications

Satisfactory graph partition, variants, and generalizations

Satisfactory graph partition, variants, and generalizations

Long Local Searches for Maximal Bipartite Subgraphs

A Max-Cut approximation using a graph based MBO scheme

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